The space of $L^2$ functions on the unit sphere $S\subset {\mathbb{R}}^n$ with the round metric is a direct sum $\oplus_{m=0}^\infty H_m(S)$ where $H_m(S)$ is the vector space of harmonic homogeneous polynomials of degree $m$ restricted to the sphere.

Let $p$ be a harmonic homogenous polynomial of even degree $2d$. If $p$ is positive on the sphere, is $p$ a sum of squares of $L^2$ functions on the sphere? That is, can we find $g_1,\ldots, g_k\in L^2(S)$ such that $p=g_1^2+\cdots g_k^2$?

A theorem of Reznick states that for a homogeneous polynomial $p\in {\mathbb{R}}[x_1,\cdots, x_n]$ that is positive on the sphere $S$, then there exists large $N$ such that $|x|^{2N}p$ is a sum of squares of polynomials. Here $|x|^2=x_1^2+\cdots x_n^2$ is the standard Euclidean norm. Since $|x|\equiv 1$ on $S$, therefore my question amounts to asking whether the sums of squares can be chosen to be from $L^2(S)$.

The geometric interpretation is that the map $G=(g_1,\ldots, g_k):S\to {\mathbb{R}}^k$ is an $L^2$ isometry in the sense that $G^* |\bullet |=p$.